What is a lappet, you might reasonably ask. And the answer would change, depending on whether you ask a biologist, a student of ecclesiastical garb, or a vintage fashion fan.
Let's get the biologists out of the way first, for in their world lappets are loose folds or flaps in an animal's anatomy--for example the wattles on a turkey's neck, or the ruffles trailing down from a jellyfish's bell.
In church, a lappet is a band of cloth or lace that is attached to the inside of ceremonial tiara or mitre of a pope or bishop, with ends left to hang down the back. According to Wiki, it is believed that these were originally devised as a kind of sweatband, lending added dignity to the proceedings during the scorching Vatican City summers.
For our purposes, a lappet is a length of material, quite often gorgeously intricate lace, that might have been attached to a cap, or worn as a collar. The intricacy of the handwork would bring additional splendor to a costume already encrusted with lace, or might act on its own as the sole bit of frippery in an otherwise dour outfit of black and white.
If you are lucky enough to find a vintage lappet like the one at top in good order for a reasonable price, it would make one of the most brilliant summer scarves going: airy enough to wear on warm days, and absolutely gorgeous against bare skin. A lappet of sufficient length and strength could also act as a wonderful belt for a linen dress. Old lacework can be worn in such lovely ways, and it's nearly always underpriced, particularly at vintage and antique fairs.